A number of systems and programs are offered on the market for the design of parts or assemblies of parts, such as the one provided by the applicant under the trademark CATIA. These so-called computer-aided design (CAD) systems allow a user to construct and manipulate complex three-dimensional (3D) models of parts or assemblies of parts. These systems and programs use various constraints for defining models. The set of constraints is solved by the system when the model is edited. The program or system used for solving the set of constraints is generally called a “solver”. Such solvers are used in CAD/CAM/CAE systems or more generally in any system using constraints for defining objects of any kind. A solver is adapted to the objects being designed and to the type of constraints applied to these objects. A solver such as the one used in CATIA is adapted for the design of solid objects, with constraints comprising dimensional constraints, stresses, contacts between objects and the like.
There is a need for a solver making it possible to simulate the deformed shapes of slender-body flexible solid physical systems such as cables, hoses, tubes, pipes, belts, foils, gaitors, ribbons, harnesses, chains, wires, ropes, strings, beams, rods, shafts, springs, etc, alone or in combination. Usual solvers are not always adapted to describing the constraints of such slender-body flexible systems; compared to objects usually designed in CAD/CAM/CAE systems, slender-body flexible systems are characterized by the fact that their aspect ratio (AR=largest dimension/mean value of smaller dimensions, i.e.: length/mean cross-section diameter) is typically high (e.g.: AR>5); in other words, their shape can be described by a curve (“neutral line”) along with transverse cross-section characteristics. This feature is representative of the fact that the system consists of slender bodies.
Another difference between such slender-body flexible systems and the objects usually designed in CAD/CAM/CAE systems is their high deformability (“geometrical non-linearity”) in the sense that distinct points on the neutral line can independently undergo arbitrarily large rotations under the action of loads. In other words, for slender-body flexible systems, the deformed shape of the system may substantially differ from the non-deformed shape of the system. On the contrary, geometrically linear solvers in existing CAD/CAM/CAE systems work under the assumption that the deformed shape of the object is close to the non-deformed shape. More generally, systems undergoing large rotations, such that the rotation angles in radians cannot be approximated by the tangent—typically 5 degrees, but this figure may vary depending on the amount of modeling error tolerated—are usually considered as “geometrically non-linear systems”. Alternatively, one could say that a system is non-linear when a “large strain” definition such as the Green-Lagrange measure:
      ɛ    ij    GL    =            1      2        ⁡          [                                    ∂                          u              i                                            ∂                          X              j                                      +                              ∂                          u              j                                            ∂                          X              i                                      +                                            ∂                              u                k                                                    ∂                              X                i                                              ⁢                                    ∂                              u                k                                                    ∂                              X                j                                                        ]      cannot be replaced by the “small strain” definition of the strain tensor components:
      ɛ    ij    LIN    =            1      2        ⁢          (                                    ∂                          u              i                                            ∂                          X              j                                      +                              ∂                          u              j                                            ∂                          X              i                                          )      that is when the product term in the Green-Lagrange measure
            ∂              u        k                    ∂              X        i              ⁢          ⁢            ∂              u        k                    ∂              X        j            cannot be neglected
The need for a solver adapted to slender-body flexible systems is particularly present in automotive and aerospace industries. In both industries, there is a need to design and simulate cables, pipes etc. throughout the body of the vehicle or plane.
Use of the FEM for the Design of Flexible Parts, Charles-André de Hillerin, Proceedings of NAFEMS World Congress 1999, pp. 345-356, discusses simulating of the behavior of highly flexible components by using integrated CAD-CAE tools. This document describes a method for computing the equilibrium shapes of a flexible cable of arbitrary cross-section, subjected to prescribed end positions and orientations, based on an incremental total Lagrangian formulation, with controlled co-rotational updating. The solution is obtained with a direct gradient method by performing an exact line search at each iteration.
This document does not teach how to carry out the co-rotational updating in the incremental total Lagrangian formulation. In addition, this document only encompasses prescribed end positions, and does not contemplate any release of the degree of freedom of the system at the end positions. Last, the starting cable configuration in this document is a rectilinear rest shape of the flexible elongated system.
Y. Toi et al, Finite element of superelastic, large deformation behaviour of shape memory alloy helical springs, Computers and Structures 82 (2004), 1685-1693, discusses a finite element analysis, using a total Lagrangian approach. There is no reference whatsoever in this document of a co-rotational update. Indeed, in page 1688, right column, section 3.2, this document indicates that the non-linear terms with respect to the displacement in the axial direction are neglected. This indicates that non-linear terms for the rotational displacements will not be neglected; otherwise, the problem would become a totally linear problem, which would not require the non-linear iterative computation discussed in the document. In view of this, co-rotational update need not be carried out.